ADDITIONS TO A MEMOIR ON METHODS OF INTERPOLATION AP- 

 PLICABLE TO THE GRADUATION OF IRREGULAR SERIES. 



By E. L. De Forest, M. A. 



The memoir referred to may be found at page 275 of the appendix to 

 the Smithsonian Eeport of 1871. Its contents need not be repeated 

 liere, and the reader will be presumed to have that report before him. 

 The formulas and tables we shall now obtain will be numbered consecu- 

 tively with the previous ones. 



INTEEPOLATION BY MEANS OF AN ALGEBRAIC FUNCTION. — FIRST 



METHOD. 



The question was left unsettled whether it is best to assume groups 

 of equal extent, as in the case of formulas A, B, 0, &c., (pp. 279 to 285,) 

 or to make them of unequal extent and in accordance with Tchebicheff's 

 system of arrangement, as in formulas (40), (41), and (42). Some further 

 investigation has made it seem probable that the latter system, though 

 not always the best, is the most likely to give uniformly good results. It 

 agrees better with Cauchy's method of interpolation, and compares very 

 favorably with that method, so far as can be judged from a few trials 

 which have been made. Hence it is desirable to extend the series of 

 formulas (40), (41), &c., so as to include the cases of six, seven, eight, and 

 nine assumed groups. This has been done, and the numerical co-effl- 

 cients involved have been carried out to as many as nine significant 

 figures — a larger number than is required in ordinary i^ractice 5 but it 

 was thought best to compute them, once for all, with as great accuracy 

 as can ever be needed for any purpose. The labor of computation need 

 never be undertaken again, for their accuracy can be easily tested, as 

 follows. Take, for instance, formula (42), and suppose that the terms of 

 the given scries are each equal to unity; then in the equation of this 

 series we ought to have A = 1, while all the other constants, B, C, &c., 

 should be zero. The sums Si, S2, S3, &c., are equal to Wi, %, %, &c.^ 

 respectively, so that we have — 



A = 3.777709 x .3090170 + | x .0954915 - .4111456 x^ = 1.0000001 

 This differs from unity by only an unit of the seventh decimal place, 

 which is as close an approach to exactness as can be attained without 

 carrying out the decimals farther than is done in formula (42). So, too, 

 in the cases of the constants C and E, we have — 



1 2SS 1 



C =2^,(55.33375 x ^ - 71.73251 x .3090170 - ^x .0954915) =^^{ ~ .0000004) 



E=?S^(.3090170+2x. 0954915 -*)=0 



